Amy is now (year 0) 25 years old and planning for her retirement at age 65. She would like to have $2 million for her retirement by then. She will make her first deposit in her retirement savings account next year (year 1) and continue until her last deposit at age 65 (year 40). All deposits are made on the last day of the year. Amy expects her income to increase at an annual rate of 2.64% (nominal) and thus will increase her savings at the same rate. Her savings will earn an annual return of 6.73% (nominal).  Suppose that inflation is 2.64% per year   Calulate intial year 0 deposit

To calculate the initial deposit Amy needs to make in year 0 to reach her goal of $2 million by age 65, we'll consider the increasing series of deposits, the rate of return on the savings, and the effect of inflation. Here’s how we’ll approach the problem:

### Step 1: Analyze the given information
- **Amy's age now (year 0)**: 25 years
- **Retirement age (year 40)**: 65 years
- **Final amount desired**: $2,000,000
- **Annual savings growth rate**: 2.64% (nominal)
- **Annual return rate on savings**: 6.73% (nominal)
- **Inflation rate**: 2.64% per year

### Step 2: Identify the required formula
Since Amy will increase her savings by a constant nominal growth rate and the savings will accumulate with interest, we can use the formula for the future value of a growing annuity:

\[
FV = P \times \frac{(1 + r)^n - (1 + g)^n}{r - g}
\]

Where:
- $ FV $ is the future value of the annuity (in this case, $2,000,000),
- $ P $ is the first deposit amount (what we need to find),
- $ r $ is the annual return rate (6.73%),
- $ g $ is the annual growth rate of the deposits (2.64%),
- $ n $ is the number of periods (40 years).

### Step 3: Solve for the initial deposit $ P $

Rearrange the formula to solve for $ P $:

\[
P = \frac{FV \times (r - g)}{(1 + r)^n - (1 + g)^n}
\]

Now, let’s plug in the values:

\[
r = 0.0673
\]
\[
g = 0.0264
\]
\[
n = 40
\]
\[
FV = 2,000,000
\]

### Step 4: Perform the calculation

\[
P = \frac{2,000,000 \times (0.0673 - 0.0264)}{(1 + 0.0673)^{40} - (1 + 0.0264)^{40}}
\]

Now, let's calculate the value.The initial deposit Amy needs to make at the end of year 1 is approximately **$7,645.34**. By increasing her savings each year by 2.64% and earning an annual return of 6.73%, she will accumulate $2 million by the time she retires at age 65.

Would you like to dive into more details or ask any questions? Here are some related questions to consider:

1. How does changing the expected return rate affect the initial deposit required?
2. What if Amy could save more or less each year—how would that impact her retirement goal?
3. How does inflation affect the real value of Amy's savings over time?
4. How would delaying her first deposit by one year impact the required savings?
5. What if the retirement goal is adjusted for inflation? How does that change the calculation?

**Tip:** When planning for retirement, it's crucial to consider both inflation and investment returns to ensure your savings maintain their value over time.

Learn how to calculate the initial deposit Amy needs to make for her retirement savings plan, considering annual savings growth, investment returns, and inflation. Follow a detailed step-by-step solution with formulas and numerical calculations.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation